Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,32)(28,34)(29,39)(30,40)(31,33)(35,38)(36,37);(3,7)(6,8)(10,14)(15,17)(16,20)(22,23)(25,27)(26,32)(28,31)(29,35)(30,36)(33,34)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,20)(19,24)(26,33)(27,28)(29,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,27)(26,34,33,32)(29,37,40,35)(30,38,39,36);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29)(26,30)(27,35)(28,37)(31,40)(32,36)(33,39)(34,38);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30)(26,29)(27,36)(28,38)(31,39)(32,35)(33,40)(34,37)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034853);